DECEMBER 2005
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MÖLDERS ET AL.
Application of Gaussian Error Propagation Principles for Theoretical
Assessment of Model Uncertainty in Simulated Soil Processes Caused by
Thermal and Hydraulic Parameters
NICOLE MÖLDERS, MIHAILO JANKOV,
AND
GERHARD KRAMM
Geophysical Institute, University of Alaska Fairbanks, Fairbanks, Alaska
(Manuscript received 29 July 2004, in final form 31 May 2005)
ABSTRACT
Statistical uncertainty in soil temperature and volumetric water content and related moisture and heat
fluxes predicted by a state-of-the-art soil module [embedded in a numerical weather prediction (NWP)
model] is analyzed by Gaussian error-propagation (GEP) principles. This kind of uncertainty results from
the indispensable use of empirical soil parameters. Since for the same thermodynamic and hydrological
surface forcing and mean empirical parameters a soil module always provides the same mean value and
standard deviation, uncertainty is first theoretically analyzed using artificial data for a wide range of soil
conditions. Second, NWP results obtained for Alaska during a July episode are elucidated in relation to the
authors’ theoretical findings.
It is shown that uncertainty in predicted soil temperature and volumetric water content is of minor
importance except during phase transition. Then the freeze–thaw term dominates and leads to soil temperature and moisture uncertainties of more than 15.8 K and 0.212 m3 m⫺3 in mineral soils. Heat-flux
uncertainty is of the same order of magnitude as typical errors in soil-heat-flux measurements.
Uncertainty in the pore-size distribution index dominates uncertainty for all state variables and soil fluxes
under most conditions. Uncertainties in hydraulic parameters (saturated hydraulic conductivity, pore-size
distribution index, porosity, saturated water potential) affect soil-temperature uncertainty more than those
in thermal parameters (density and specific heat capacity of dry soil material). Based on a thermal conductivity approach alternatively used, it is demonstrated that GEP principles are indispensable for evaluating parameterized soil-transfer processes.
Generally, statistical uncertainty decreases with depth. Close beneath the surface, the uncertainty in
predicted soil temperature, volumetric water content, and soil-moisture and heat fluxes undergoes a diurnal
cycle.
1. Introduction
All state-of-the-art numerical weather prediction
models (NWPMs) and general circulation models
(GCMs) use soil models embedded in so-called land
surface models (LSMs) to predict the lower boundary
conditions (thermodynamic and hydrological surface
forcing), that is, temperature and specific humidity and
fluxes of water vapor and sensible heat at the soil–
atmosphere interface. These soil models have been developed based on the best knowledge of the scientific
Corresponding author address: Nicole Mölders, Geophysical Institute, University of Alaska Fairbanks, 903 Koyukuk Drive, P.O.
Box 757320, Fairbanks, AK 99775-7320.
E-mail: molders@gi.alaska.edu
© 2005 American Meteorological Society
community, and great efforts have been made to evaluate and improve them (e.g., Yang et al. 1995; Shao and
Henderson-Sellers 1996; Lohmann et al. 1998; Mölders
et al. 2003a). However, incomplete knowledge of initial
conditions and soil type and heterogeneity may generally reduce the predictability of the soil state, fluxes of
heat, water vapor, and water, and phase-transition processes within the soil and at the soil–atmosphere interface. The same is true when imperfect parameterizations for subgrid-scale processes, surface runoff, and
cloud microphysical processes lead to an erroneous
thermodynamic and hydrological surface forcing. Since
NWPMs and GCMs are deterministic in nature, we
must recognize that these shortcomings are systematic
(or procedural) errors. Such errors, of course, can cause
unacceptably great uncertainty in predicted results. Un-
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JOURNAL OF HYDROMETEOROLOGY
fortunately, this uncertainty cannot be evaluated using
statistical methods because procedural errors must generally be removed or, at least, minimized before statistical methods can be employed (e.g., Kreyszig 1970).
Much work has focused on identifying such procedural errors in the thermodynamic and hydrological
surface forcing and evaluating the uncertainty they
cause:
(i) Uncertainty in the quantities at the surface that
results from initializing soil-moisture and temperature distributions has already been investigated for
various stand-alone versions of LSMs (e.g., Gao et
al. 1996), NWPMs (e.g., Douville and Chauvin
2000), and GCMs (e.g., Wang and Kumar 1998).
Adjoint models and data-assimilation techniques
also have been applied for minimizing errors in
initial soil conditions (e.g., van den Hurk et al.
1997; Callies et al. 1998; Reichle et al. 2001).
(ii) Various sensitivity studies were performed to detect error sources related to presumptions and/or
parameterization concepts (e.g., Robock et al.
1995; Cuenca et al. 1996; Shao and Irannejad
1999). The force-restore method, for instance, as it
works with two or three reservoirs, has only limited ability to resolve soil horizons (Montaldo and
Albertson 2001) and to simulate the vertical distributions of soil processes like the diurnal variation of the freezing line.
(iii) Uncertainties related to parameterization of subgrid-scale processes, surface runoff, and cloud microphysical processes including formation of precipitation, precipitation interception on different
vegetation, and contribution of precipitation to
thermodynamic and hydrological surface forcing
have been evaluated by various authors (e.g.,
Avissar and Pielke 1989; Calder et al. 1995;
Mölders et al. 1996, 1997; Niu and Yang 2004).
Ideally, soil characteristics are mapped as continuous distributions to capture the gradients and mixtures in soil within a grid cell. Soils, however, are
usually heterogeneous in space; consequently attributing a single soil type to an area of several
square kilometers as is done in NWPM soil models
can be ambiguous, and may surely be a source of
errors. Using a wrong soil type, for instance,
causes errors of more than 0.5 K and 0.5 g kg⫺1 for
near-surface air temperatures and humidity even
in a 24-h simulation (Mölders 2001). It is well
known that the variability in some soil parameters
(e.g., hydraulic conductivity) is sometimes greater
within the same soil type than across soil types.
Observations show that heterogeneity within the
VOLUME 6
same soil may cause differences in evapotranspiration and recharge of 112 (14%) and 137 mm
(4%) in 2050 days (Mölders et al. 2003b).
(iv) Another kind of systematic error is related to the
application of NWPMs themselves. Since the forecast range of such models is generally restricted to
a week or so, predictions that violate this limitation of integration time can cause such great uncertainty that the results become worthless. This
kind of uncertainty is well known and therefore
not further discussed here.
(v) Results from the Project for Intercomparison of
Land Surface Parameterization Schemes (PILPS)
showed that LSMs strongly differ in accuracy because of, among other things, the choice of empirical parameters required in parameterizations (e.g.,
Shao and Henderson-Sellers 1996; Slater et al.
1998). Usually, a mean value for a soil property
derived from laboratory or/and field studies is attributed to a grid element (ignoring any kind of
uncertainty). Consequently, predicted heat and
matter fluxes and temperature and moisture distributions can vary over wide ranges depending on
the choice of such empirical parameters, customarily considered as fixed during a simulation. If, for
instance, distributions of prescribed coverage of
land-use and soil types differ by about 5%, daily
averages of the soil-moisture fraction alter by 0.19,
a 29% change with respect to the reference case,
and surface temperature by 2.3 K (Mölders et al.
1997). To assess whether slightly different parameters will result in significant perturbations of the
model result, many simulations are required
wherein (in the sense of parameter variation) only
one parameter is altered at a time. Such parameter-variation studies are subject to so-called parameter interaction; that is, parameter choice affects other simulated quantities indirectly. Parameter effects or parameterization deficits that
accidentally cancel each other out can remain
overlooked (Henderson-Sellers 1993). To minimize parameter interaction these methods must be
driven by either observation or reanalysis.
The inevitable use of empirical parameters in parameterizations for describing transfer processes leads to
another source of uncertainty. Empirical parameters
are generally burdened with “errors” arising from natural (random) variability as expressed by the variance or
standard deviation. These standard deviations can be of
the same order of magnitude as the parameters themselves (e.g., Clapp and Hornberger 1978; Cosby et al.
1984). Consequently, any quantity predicted with these
DECEMBER 2005
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MÖLDERS ET AL.
parameters is “error” burdened. Such uncertainty may
even reduce the trust in short-term NWP and, in particular, the gain of credible information for agricultural
use. In the case of climate simulations, this uncertainty
may be a great burden in climate impact assessment.
These errors are of random kind and, hence, can be
evaluated with statistical methods.
To systematically investigate this statistical uncertainty that accompanies the predicted distributions of
volumetric water content, soil temperature, and moisture and heat fluxes because of empirical soil parameters, we introduce Gaussian error-propagation (GEP)
principles. The aim is to identify critical parameters (to
prioritize which quantities to measure with higher accuracy), to point out parameterizations that cause high
uncertainty due to their parameter dependency and the
inherent statistical uncertainty, and to find possibilities
for soil-modeling improvements. Since in a mathematical sense an equation to predict a flux/state variable and
calculate the standard deviation is unambiguous, it will
always provide the same flux/state variable and standard deviation for the same set of state variables and
empirical parameters. Therefore, the statistical uncertainty of predicted soil conditions is first analyzed using
artificial data for the typical range of thermodynamic
and hydrological surface forcing. Doing so permits discussion of the spectrum from soil frost to relatively
warm soil, and from dry to wet soil conditions. Second,
to provide an example we assess the meaning of the
theoretical findings for modeling by quantifying the
model uncertainty throughout an NWP for Alaska during a summer episode.
ized according to Hong and Pan (1996). The hydrothermodynamic soil–vegetation scheme (HTSVS)
serves to determine the lower boundary conditions of
MM5, that is, temperature and specific humidity and
exchange of momentum, heat, and moisture at the
earth–atmosphere interface. HTSVS consists of a multilayer soil model, a single-layer canopy model (Kramm
et al. 1994, 1996; Mölders et al. 2003a), and a multilayer
snow model (Mölders and Walsh 2004). Long-term
(2050 days) evaluations demonstrated that HTSVS
simulates accumulated groundwater recharge, evapotranspiration, and soil temperatures within 15% and on
average about 1–2-K accuracy without calibration
(Mölders et al. 2003a,b).
b. Soil model
The soil model is based on the principles of the linear
thermodynamics of irreversible processes (e.g., de
Groot 1951). It deals with soil freezing/thawing and
(vertical) heat- and water-transfer processes (including
the Richards equation) quantified by the balance equations for heat and moisture (e.g., Philip and de Vries
1957; de Vries 1958; Kramm et al. 1994, 1996; Mölders
et al. 2003a),
C
⫹
⫹
a. Model setup
冊
⭸
⭸␩
⭸␩ice
,
L␷␳wD␩␷
⫹ Lf ␳ice
⭸zS
⭸zS
⭸t
⭸
⭸
⭸␩
⭸␩
⭸␩
⫽
D␩␷
⫹
D␩ l
⭸t ⭸zS
⭸zS
⭸zS
⭸zS
2. Description of the NWP model
The fifth-generation Pennsylvania State University–
National Center for Atmospheric Research (NCAR)
Mesoscale Model (MM5) is the NWPM used in this
study. As MM5 has evolved over the last decade and
has been thoroughly documented and widely used (e.g.,
Dudhia 1993; Chen and Dudhia 2001), the model setup
is only briefly described.
Cloud microphysical processes are predicted by Reisner et al.’s (1998) mixed-phase scheme that distinguishes between cloud water, rainwater, ice, snow, and
graupel. As the horizontal resolution (45 km) considerably exceeds the typical horizontal extension of cumulus clouds, a cumulus parameterization is inevitable
(e.g., Raymond and Emanuel 1993), for which we use
Grell et al.’s (1991) cumulus scheme. Additionally,
Grell et al.’s (1994) simple radiation scheme is considered. The turbulent transfer processes are parameter-
冉 冊 冉
冉
冊
冉 冊 冉
冉 冊
⭸
⭸
⭸TS
⭸TS
⭸TS
⫽
␭
⫹
L ␷ ␳w D T ␷
⭸t
⭸zS
⭸zS
⭸zS
⭸zS
共1兲
冊
⭸Kw
⭸
⭸TS
␹
␳ice ⭸␩ice
.
DT ␷
⫹
⫺
⫺
⭸zS
⭸zS
⭸zS
␳w
␳w ⭸t
共2兲
Here, zS is soil depth, and ␹ represents water uptake
per soil volume by roots;
D␩␷ ⫽ ⫺␣Dwb
␩s ⫺ ␩ ␳ a g␺
,
␩ ␳w R d T S
共3兲
␳a L␷ ⫺ g ␺
,
␳w RdT S2
共4兲
DT␷ ⫽ ␣Dw共␩s ⫺ ␩兲
and
D ␩l ⫽ ⫺
冉冊
bks␺s ␩
␩
␩s
b⫹3
共5兲
are transfer coefficients for water vapor, heat, and water with ␣, Dw, ␩s, b, ␺s, ks, and Rd being torsion factor,
molecular diffusion coefficient of water vapor, porosity,
pore-size distribution index, saturated water potential,
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JOURNAL OF HYDROMETEOROLOGY
saturated hydraulic conductivity, and dry air gas constant, respectively. The variables TS, ␩, and ␩ice are soil
temperature, volumetric water, and ice content. Furthermore,
K w ⫽ ksW
2b⫹3
共6兲
is hydraulic conductivity, where W ⫽ ␩/␩s is relative
volumetric water content (e.g., Clapp and Hornberger
1978). Volumetric heat capacity of moist soil is given by
(e.g., Mölders et al. 2003a)
␭⫽
再
␺⫽
冦
␺sW⫺b
Lf
Ts ⫺ 273.15
gTs
再
⫹ 共␩s ⫺ ␩ ⫺ ␩ice兲␳acp,
0.172
2 ⫹ 10logⱍ␺ⱍ ⱖ 5.1
Ts ⬎ 273.15 K
Ts ⱕ 273.15 K
,
共9兲
Lf 共TS ⫺ 273.15兲
g␺sT S
冎
⫺1Ⲑb
.
共10兲
共7兲
where ␳S, ␳w, and ␳ice are densities of dry soil material,
water, and ice; cS, cw, and, cice are the corresponding
specific heats. Moreover, ␳a and cp are the density of air
and specific heat at constant pressure; L␷ and Lf are the
latent heats of vaporization and freezing.
In accord with McCumber and Pielke (1981), thermal
conductivity,
2 ⫹ 10logⱍ␺ⱍ ⬍ 5.1
where g is acceleration due to gravity. Below 273.15 K,
mass-weighted thermal conductivity depending on
volumetric water and ice content is calculated using Eq.
(8) for the liquid and 2.31 J (m s K)⫺1 for the solid
phase.
In Eq. (1), the first term on the right-hand side addresses soil-temperature changes by divergence of soilheat fluxes. The second term denotes the divergence of
soil-heat fluxes due to water vapor transfer, the third
refers to the Dufour effect (i.e., a moisture gradient
contributes to a soil-temperature change), and the last
term gives changes due to freezing/thawing. The first
two terms on the right-hand side of Eq. (2) give the
changes in volumetric water content due to divergence
of water vapor and water fluxes. The third term represents the Ludwig–Soret effect (i.e., a temperature gradient contributes to a volumetric water-content
change), the fourth denotes changes due to hydraulic
conductivity, the fifth is water uptake by roots, and the
last describes freezing/thawing. The Dufour and Ludwig–Soret effects are usually considered cross phenomena in the thermodynamics of irreversible processes.
At temperatures below freezing, water and ice can
coexist in soils. The maximum amount of supercooled
water that may exist is given by (e.g., Flerchinger and
Saxton 1989)
␩max ⫽ ␩s
C ⫽ 共1 ⫺ ␩s兲␳ScS ⫹ ␩␳wcw ⫹ ␩ice␳icecice
419 exp兵⫺共2 ⫹ 10logⱍ␺ⱍ ⫹ 2.7兲其
depends on water potential (e.g., Flerchinger and Saxton 1989)
VOLUME 6
,
共8兲
As most state-of-the-art soil models (e.g., Sievers et
al. 1987; Verseghy 1991; Viterbo and Beljaars 1995;
Wetzel and Boone 1995; Yang et al. 1997; Slater et al.
1998; Viterbo et al. 1999; Chen and Dudhia 2001; Warrach et al. 2001) are based on variations or simplified
versions of Eqs. (1)–(10), the following uncertainty
analysis of their terms permits generalizing.
c. Model domain
The model domain (Fig. 1) has 39 ⫻ 39 points, a
horizontal grid spacing of 45 km, and 23 vertical layers
reaching to 100 hPa. There are five snow layers of equal
thickness depending on snow depth. The uppermost
FIG. 1. Soil-type distribution and topography as used in the
study.
DECEMBER 2005
1049
MÖLDERS ET AL.
soil layer ranges from the surface to the uppermost
level within the soil at 0.1-m depth. Between that level
and the lowest level at 2.95-m depth, there are four
layers spaced by the same logarithmic increment so that
central differences can be used in solving the coupled
Eqs. (1) and (2) by a generalized Crank–Nicholson
scheme. The time step is 135 s.
d. Synoptic situation
The episode covers 0000 UTC 20 July to 1200 UTC
23 July 2001. During this period, weather was driven by
cyclonal activity within a large-scale trough over the
North Pacific and Bering Sea. The long-wave disturbance propagated eastward and established a weakgradient spacious trough over Alaska in the middle and
toward the end of the episode. On the surface, Alaska
remained on the outer edge of a well-developed eastward-moving cyclone located north of Siberia. This synoptic pattern was favorable for heavy thunderstorms
over the Alaska and Brooks Ranges and the Interior.
During the episode, low-elevation near-surface air and
dewpoint temperatures ranged from 8° to 22°C and ⫺1°
to 16°C, respectively. Sunrise (sunset) is around 0300
(2100), 0400 (2330), and 0530 (0100) Alaska standard
time (AST; UTC minus 9 h) in the eastern, central, and
western parts of the domain. While the sun barely goes
below the horizon in the north, sunrise (sunset) is
around 0500 (2300) AST in the southern central part of
the domain.
e. Initialization
Initial and boundary conditions are obtained from
the National Centers for Environmental Prediction
(NECP)–NCAR reanalysis project (NNRP). The vegetation fraction of each grid cell is a weighted combination of July and August monthly 5-yr-mean green
vegetation cover (0.15° resolution) derived from Advanced Very High Resolution Radiometer (AVHRR)
data (Gutman and Ignatov 1998). Soil texture, terrain
elevation (Fig. 1), and land use are taken from the
1-km-resolution U.S. Department of Agriculture
(USDA) State Soil Geographic Database (Miller and
White 1998) and 10-min-resolution U.S. Geological
Survey (USGS) terrain and vegetation data.
Initial total soil moisture and temperature are interpolated from NNRP data. Total moisture partitioning
between liquid and solid phases follows Mölders and
Walsh (2004). No assumptions about active layer depth
are required as HTSVS works for both active layer and
permafrost. Temperature, volumetric water, and ice
content at the bottom of the soil model are constant
throughout the simulation.
3. Method
a. Gaussian error propagation
The state quantities ␩(␩s, ␺s, b, ks) and Ts(␩s, ks, b,
␳s, cs) and the related moisture Ws(␩s, ␺s, b, ks) and heat
Hs(␩s, ␺s, b, ␳s, cs) fluxes are “error” burdened by an
amount ␴␾ resulting from the random variability of empirical parameters usually characterized by standard
deviations. To determine model uncertainty caused by
empirical soil parameters for ␩, Ts, Ws, and Hs at the
surface and various depths, we consider GEP principles. In doing so, the equation to predict a quantity ␾
(e.g., ␩) is derived for all empirical parameters ␹i on
which it depends (e.g., ␩s, ␺s, b, and ks). The standard
deviation (statistical uncertainty) of the predicted
quantity can be calculated from these individual derivations (⳵␾/⳵␹i) and the standard deviations ␴␹i of the
ith empirical parameters ␹i by (e.g., Kreyszig 1970)
冑兺 冉
n
␴␾ ⫽
i⫽1
⭸␾
⭸␹i
冊
2
冑兺
n
␴␹2 i ⫽
i⫽1
兵␾, ␴␹i 其2, 共11兲
where n is the number of parameters, ␴2␹i are the variances, and (⳵␾/⳵␹i)␴␹i :⫽ {␾, ␴␹i} and ␴␾ are denoted
contribution (term) and uncertainty, hereafter. The
relative error is defined by ␧␾ ⫽ (␴␾ /␾). Note that one
standard deviation means that 68.27% of all values fall
within ␾ ⫾ ␴␾.
Equation (11) assumes that 1) errors are normal distributed and 2) errors are independent between various
model parameters, which is justified for these parameters. Standard deviations of ␩s, ␺s, b, and ks are taken
from Clapp and Hornberger (1978) and Cosby et al.
(1984) for mineral soils. Standard deviations for organic
soils and ␳S, and cS are collected from various sources
(Table 1). To our best knowledge there are no studies
on the sensitivity of the parameter variance at different
temperature and moisture conditions. Therefore, as
common practice in soil modeling, we assume these
parameters and their variances as independent of the
state variables.
b. Uncertainty analysis
According to Eq. (11), any set of mean parameters
and their standard deviations always provide the same
standard deviation of a predicted quantity for the same
soil conditions. Therefore, equations for ␩, Ts, Ws, and
Hs and the respective equations for standard deviation
are applied for typical soil-forcing ranges. Fluxes and
their standard deviations are calculated for soil temperatures and volumetric water content from ⫺20° to
30°C and 0.001 to porosity, respectively. Soil-moisture
0.339 ⫾ 0.073 (21.53)
0.421 ⫾ 0.072 (17.10)
0.474 ⫾ 0.088 (1.69)
0.434 ⫾ 0.054 (12.44)
0.439 ⫾ 0.074 (16.86)
0.404 ⫾ 0.048 (11.88)
0.464 ⫾ 0.046 (9.91)
0.465 ⫾ 0.054 (11.61)
0.406 ⫾ 0.032 (7.88)
0.468 ⫾ 0.062 (13.25)
0.468 ⫾ 0.035 (7.48)
0.923 ⫾ 0.342 (37.05)
0.900 ⫾ 0.040 (4.44)
0.95 ⫾ 0.060 (6.67)
46.8 ⫾ 16.29 (34.81)
14.19 ⫾ 23.7 (167.02)
5.27 ⫾ 1.52 (28.84)
2.83 ⫾ 2.00 (70.67)
3.38 ⫾ 1.67 (49.41)
4.487 ⫾ 2.05 (45.69)
2.051 ⫾ 1.75 (85.32)
2.466 ⫾ 1.83 (74.21)
7.28 ⫾ 1.50 (20.60)
1.355 ⫾ 1.45 (100.70)
0.9816 ⫾ 1.705 (170.37)
1.736 ⫾ 0.938 (54.03)c
150 ⫾ 400 (266.67)d
3356.5 ⫾ 200 (5.96)e
Sand
Loamy sand
Sandy loam
Silt loam
Loam
Sandy clay loam
Silty clay loam
Clay loam
Sandy clay
Silty clay
Clay
Peat soil
Moss soil
Lichen soil
b
Grunwald et al. (2001).
Calhoun et al. (2001).
c
Schlotzhauer and Price (1999).
d
Carey and Woo (1999).
e
Laurén and Heiskannen (1997).
a
␩s (m3 m⫺3)
ks 10⫺6 (m s⫺1)
Soil type
2.79 ⫾ 1.38 (49.46)
4.26 ⫾ 1.95 (45.77)
4.74 ⫾ 1.40 (29.53)
5.33 ⫾ 1.72 (32.27)
5.25 ⫾ 1.66 (31.62)
6.77 ⫾ 3.39 (50.07)
8.72 ⫾ 4.33 (49.66)
8.17 ⫾ 3.74 (45.78)
10.73 ⫾ 1.54 (14.35)
10.39 ⫾ 4.27 (41.10)
11.55 ⫾ 3.93 (34.03)
4.00 ⫾ 1.75 (43.75)
1.00 ⫾ 1.75 (175)
0.50 ⫾ 1.75 (350)
b -.⫺0.069 ⫾ 0.036 (52.17)
⫺0.0361 ⫾ 0.0537 (149)
⫺0.141 ⫾ 0.0537 (38.09)
⫺0.759 ⫾ 0.024 (3.16)
⫺0.355 ⫾ 0.0457 (12.87)
⫺0.135 ⫾ 0.11 (81.48)
⫺0.617 ⫾ 0.038 (6.16)
⫺0.263 ⫾ 0.055 (20.91)
⫺0.098 ⫾ 0.0363 (37.04)
⫺0.324 ⫾ 0.069 (21.30)
⫺0.468 ⫾ 0.039 (8.33)
⫺0.165 ⫾ 0.31 (188)
⫺0.120 ⫾ 0.310 (258)
⫺0.085 ⫾ 0.310 (365)
␺s (m)
cS [J (kg K)⫺1]
930 ⫾ 76 (8.17)
876 ⫾ 69 (7.88)
882 ⫾ 34 (3.85)
907 ⫾ 69 (7.61)
896 ⫾ 52 (5.80)
776 ⫾ 75 (9.66)
936 ⫾ 85 (9.08)
866 ⫾ 72 (8.31)
783 ⫾ 48 (6.13)
797 ⫾ 52 (6.52)
890 ⫾ 23 (2.58)
1920 ⫾ 100 (5.21)
10 000 ⫾ 100 (1)
8333 ⫾ 100 (1.2)
␳S (kg m⫺3)
1580 ⫾ 90 (5.70)a
1610 ⫾ 100(6.21)a
1520 ⫾ 140(9.21)a
1400 ⫾ 90 (6.43)a
1350 ⫾ 110 (8.15)a
1520 ⫾ 40 (2.63)a
1410 ⫾ 60 (4.26)
1420 ⫾ 80 (5.63)a
1570 ⫾ 120 (7.64)a
1480 ⫾ 110 (7.43)a
1470 ⫾ 140 (9.52)b
106 ⫾ 243 (229)
100 ⫾ 100 (100)
120 ⫾ 30 (25.00)
TABLE 1. Soil characteristics for soils considered in this study. Here, ks, ␩s, b, ␺s, cS, and ␳S are the saturated hydraulic conductivity, porosity (saturated volumetric water content),
pore-size distribution index, saturated water potential, as well as the specific heat capacity and density of the dry soil material. Parameters plus/minus their standard deviations for ks,
␩s, b, and ␺s are from Cosby et al. (1984). Values in parentheses denote the relative error in %.
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MÖLDERS ET AL.
and temperature gradients are varied from ⫺200 to 200
m3 m⫺3 (m⫺1) and ⫺200 to 200 K m⫺1.
To identify critical parameters we estimate the contributions of individual parameters to the uncertainty
by analyzing the various terms {␾, ␴␹i}. Ideally all terms
{␾, ␴␹i} are of the same order of magnitude for a good
parameterization and parameter set. A parameter ␹k
will be identified as critical if {␾, ␴␹k} exceeds all other
terms {␾, ␴␹i}i⫽k by more than an order of magnitude. A
parameterization (or parts of it) will be classified as
critical if its dependent parameters lead to a huge standard deviation in this parameterization, but do not provide great uncertainty in another parameterization of
the soil model.
As mentioned before, an MM5 simulation is performed for Alaska during a summer period to demonstrate the meaning of the theoretical results for NWP
and to elucidate the limit of predictability. For these
purposes, the uncertainty-analysis tool was implemented into MM5 to avoid improper data handling by
a postprocessor. This supplementary module, of course,
does not affect any simulation results. Since MM5 starts
without clouds it takes a certain amount of time until
clouds form in MM5, leading to excessive insolation
during spinup (systematic errors). Thus, the discussion
focuses on results after spinup. As common practice in
NWPMs, we assume each MM5 grid cell to be homogeneously taken by one soil type at all depths. Note that
if soil types varied horizontally and/or vertically within
a grid cell, spatial derivations would have to be included in Eq. (11) and results may become scale dependent.
4. Results
Typically, total moisture (water plus ice) increases
with depth. In permafrost, temperatures are below 0°C
for at least two consecutive years; total moisture is close
to porosity. In the active layer, the frozen water fraction
increases with depth. Desert soils are dry and hot; soils
on desert plateaus dry but cold; and soils in the Tropics
warm and wet. Moderately wet, near-0°C conditions
correspond to midlatitude winters. Snowmelt and rain
yield wet conditions at temperatures around and above
0°C, respectively.
Because of atmospheric forcing soil-temperature and
moisture gradients are greater close beneath the surface than in deeper soil layers. Close to the surface, ␩,
Ts, Ws, and Hs undergo a diurnal cycle. Rain and snowmelt (evapotranspiration) produce downward (upward)
soil-moisture fluxes.
Soils with no clay but high sand fraction permit the
lowest, and clay-containing soils the greatest, fraction
of supercooled water (Fig. 2). At 0°C, the maximum
1051
FIG. 2. Freezing characteristic curve showing the dependence of
maximum relative supercooled water content ␩max/␩s on soil temperature for selected soils. Note that relative water content ranges
between 0 and 1 and is dimensionless.
allowable supercooled water drops to 50% of porosity
for all soils except for silty clay loam, sandy clay loam,
silt clay, and clay that reach this condition at ⫺3°, ⫺2°,
⫺4°, and ⫺11°C, respectively. Supercooled water
hardly exists in moss. At ⫺11°C, the maximum allowable supercooled water falls to 25% of porosity for all
clay-containing soils; silty loam and loam reach this
value at ⫺10° and ⫺5°C; sand, loamy sand, and sandy
loam at ⫺1°C. This means organic soils freeze quickest;
clay-containing soils freeze later than those without
clay.
a. Theoretical analysis
1) OVERVIEW
Uncertainty in predicted state variables (first moments) strongly grows with increasing absolute value of
the temperature and/or moisture gradient (Figs. 3 and
4). Consequently, uncertainty is higher in soils experiencing huge changes by heating/cooling, and/or infiltration/evapotranspiration than in soils with marginal or
no changes (in the diurnal/seasonal course). Above
0°C, uncertainty is negligibly small for all soils. Below
0°C, temperature and moisture uncertainty is the greatest (lowest) in the temperature-moisture regime above
(below) the freeze–thaw curve where water and ice cannot (can) coexist under these temperature-moisture
conditions and consequently (no) phase transitions occur. Since the freeze–thaw curves differ for the various
soils, high standard deviations occur at different temperature-moisture regimes for various soils (Fig. 2). Investigation of Eqs. (1) and (2) shows that the freeze–
thaw term dominates temperature and moisture uncertainty during phase transition.
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VOLUME 6
FIG. 4. As in Fig. 3, but for soil moisture. Example shown for
sandy loam.
FIG. 3. Soil-temperature uncertainty in dependence of (a) soil
temperature and volumetric water content (example shown for
clay loam) and (b) soil-moisture and -temperature gradient (example shown for loam).
Typically, if predicted fluxes (second moments) approach zero, uncertainty skyrockets. For nonzero
fluxes, uncertainty grows with increasing absolute value
of the flux. The steepness of the increase depends on
soil type; that is, at identical temperature-moisture conditions, soil fluxes and their uncertainties differ for various soils. Generally, relative errors are greater for
fluxes than for state variables because the former depend on the gradient of the state variables.
As compared to hydraulic parameters, uncertainty in
thermal parameters of dry soil material hardly contributes to temperature and soil-heat-flux uncertainty except above the freeze–thaw curve. Here, contributions
by thermal parameters are still smaller than those by
hydraulic parameters. The reasons are manifold. The
relative errors of the thermal parameters are much
smaller than those of the hydraulic parameters (see
Table 1), and they are linear in the terms of Eqs. (1)
and (7). On the contrary, the hydraulic parameters occur in nonlinear relations in Eqs. (1)–(6) and Eqs. (8)–
(10). These findings suggest that better knowledge of
soil hydraulic parameters offers a greater potential for
reducing statistical uncertainty in soil-temperature and
heat-flux prediction than increasing the current accuracy of soil thermal quantities.
Generally, organic soils bear the greatest uncertainty
(Tables 2 and 3); this may however be an artifact, as
data for organic soils have been obtained from a wider
variety of sources than data for mineral soils. A sensitivity study performed with the standard deviations
given by Clapp and Hornberger (1978) provided results
similar to those based on Cosby et al.’s (1984) data
presented in the following. Therefore, we restrict our
discussion to the results obtained with the more recent
data.
DECEMBER 2005
TABLE 2. Soil parameters found to cause high uncertainty in the
indicated predicted soil temperature Ts , volumetric water content
␩, soil moisture Ws, and soil-heat flux Hs. Uncertainties in density
and specific heat capacity of dry soil material contribute negligibly, and saturated hydraulic conductivity contributes less than
pore-size distribution index, porosity, or saturated water potential
under most soil conditions (therefore not included in this table).
Parentheses denote quantities where the indicated parameter is
only critical under certain conditions (see text for further details).
Soil type
Porosity
Sand
Loamy sand
Sandy loam
Silt loam
Loam
Sandy clay loam
Silty clay loam
Clay loam
Sandy clay
Silty clay
Clay
Peat soil
Moss soil
Lichen soil
␩, (Ws), Hs
2) SOIL
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MÖLDERS ET AL.
Pore-size
distribution
index
Ts , (Ws)
Ts , (Ws)
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
Ts , ␩, (Ws),
(Ws)
␩, (Ws)
(Ws)
Ws
Ws
Ws
Ws
Ws
Ws
␩
(Ws)
Saturated
water
potential
(Ts)
Hs
Hs
Hs
Hs
Hs
Hs
Hs
Hs
Hs
3) SOIL
Ws
␩
TEMPERATURE
In all mineral soils, soil-temperature uncertainty remains below 0.5 K for all relative volumetric watercontent values at temperatures above 0°C (e.g., Fig. 3).
Because of different freeze–thaw curves, the temperature-moisture range of relatively dry supercooled soil
TABLE 3. Soils with on average high (H) and low (L) uncertainty for the soil temperature Ts, volumetric water content ␩,
soil-heat flux Hs, soil-moisture flux Ws, thermal conductivity ␭,
and hydraulic conductivity Kw. Uncertainty is judged with respect
to the absolute uncertainty for other soils. Note that uncertainty
for Eq. (12) is an order of magnitude less than for Eq. (8) even if
uncertainty is indicated as H.
Soil type
Ts
Sand
Loamy sand
Sandy loam
Silt loam
Loam
Sandy clay loam
Silty clay loam
Clay loam
Sandy clay
Silty clay
Clay
Peat soil
Moss soil
Lichen soil
L
L
H
␩
Hs
Ws
L
L
L
L
L
L
L
L
L
H
H
H
H
H
H
H
H
H
H
H
H
L
L
L
␭
[Eq. (8)]
H
H
H
H
H
H
H
H
H
H
H
H
H
H
␭
[Eq. (12)]
Kw
L
L
L
L
L
L
L
L
H
L
L
where soil-temperature uncertainty is low (⬍0.1 K) is
greater for clay-containing soils than other soils. This
shift toward relatively wetter conditions goes along
with a shift of the maximum standard deviations to wetter conditions.
Above the freeze–thaw curve, sandy clay and sand
have the lowest standard deviations (⬍7.5 K), followed
by sandy loam and clay (⬍10 K); in all other mineral
soils, standard deviation exceeds 10 K (e.g., Fig. 3). For
organic and mineral soils, uncertainty shows a qualitatively similar pattern. Moss is the organic soil with the
lowest (⬍5 K) soil-temperature prediction uncertainty
(Table 3).
For sand, loamy sand, and sandy loam, {Ts, ␴b} exceeds the contributions of the other terms to temperature uncertainty by an order of magnitude, on average
(Table 2). For all mineral soils, {Ts, ␴b} gains importance as the absolute value of the temperature and/or
moisture gradient grows. On average, {Ts, ␴␺s} is twice
as high as {Ts, ␴␩s} for loamy sand.
H
H
H
H
H
L
MOISTURE
Above the freeze–thaw curve, volumetric watercontent uncertainty increases with decreasing temperature until the temperature is reached where only about
25% of the volumetric water at saturation may remain
liquid (e.g., Fig. 4). Above a critical value of relative
volumetric water content ␩/␩s, which differs with soil
type, uncertainty increases with increasing ␩/␩s in the
freeze–thaw region. Above 0°C, uncertainty also grows
with increasing ␩/␩s. This means that soil-moisture prediction bears more uncertainty in wet than dry soils.
The afore-described nonlinear growth of moisture
uncertainty with increasing moisture (e.g., Fig. 4) results from nonlinear growth of water-transfer coefficients with increasing relative volumetric water content
(Fig. 5). This means that rain or meltwater can increase
moisture uncertainty by more than an order of magnitude, as soil moistens (e.g., Fig. 6).
On average, {␩, ␴b} contributes an order of magnitude more to moisture uncertainty than the other terms
for sandy loam, silt loam, loam, sandy clay loam, and
moss, and two orders of magnitude more for silty clay
loam, silty clay, and clay loam (Table 2). The dominance of {␩, ␴b} grows with increasing absolute value of
the temperature and/or moisture gradient at temperatures below 0°C. On average, for sand and moss {␩, ␴␩s}
exceeds the other terms by about an order of magnitude. This means that for the aforementioned soils b,
and for the last-mentioned soils ␩s, should be determined with higher accuracy to improve soil-moisture
predictions. For lichen, {␩, ␴␺s} is two orders of magnitude greater than {␩, ␴ks}.
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JOURNAL OF HYDROMETEOROLOGY
VOLUME 6
FIG. 5. Uncertainty in water-transfer coefficients. Note the water-transfer coefficient D␩,l depends on pore-size distribution index, saturated hydraulic conductivity, saturated water potential,
and porosity (cf. Kramm 1995).
4) SOIL-MOISTURE
FLUX
Given the same relative volumetric water-content
and temperature conditions, the various soils provide
different soil-moisture fluxes and hence uncertainty.
Soils with high clay fraction allow for the highest (lowest) soil-moisture fluxes at high (low) relative water
content ␩/␩s of all soils. Over large temperaturemoisture ranges, moisture-flux uncertainty is lower
than the flux, and uncertainty grows with increasing
absolute flux value. Uncertainty grows with increasing
␩/␩s for all soils (Fig. 7).
At temperatures above 0°C, soil-moisture fluxes and
their uncertainty marginally depend on soil temperature (Fig. 7). Only for ␩/␩s less than 0.15 the influence
of the heat-transfer coefficient, and hence temperature,
is manifested as a slight slope in the contour lines.
Fluxes and their uncertainties are smaller below than
above 0°C, and smallest below the freeze–thaw curve.
Thus, predicted soil-moisture fluxes bear more statistical uncertainty during rain, when soils are wet, or nonfrozen, than under dry or frozen ground conditions
(e.g., Fig. 8).
Obviously, all soils have a range of ␩/␩s and Ts combinations wherein differences between soil-moisture
flux and its uncertainty are maximal; that is, relative
errors are minimal. In clay loam, for instance, the lowest relative error exists at soil temperatures below 263
K and relative volumetric water content between 0.3
and 0.5; in loam, below 257 K and at ␩/␩s between 0.2
and 0.4.
On average {Ws, ␴␩s} dominates moisture-flux uncertainty for sand, sandy loam, silt loam, sandy clay loam,
clay loam, sandy clay, peat, and lichen, while {Ws, ␴␺s}
dominates for moss.
FIG. 6. Horizontal distribution of (a) volumetric water content
(contour lines) and its uncertainty (gray shading) in the second
soil layer (counted from the surface) after 45 h of simulation (m3
m⫺3), and (b) 84-h-accumulated precipitation (lines) and layer of
the freezing line (shaded) at start of simulation.
Examination of the Eq. (2) terms shows that sensitivity of the water-transfer coefficient to uncertainty in
pore-size distribution index dominates moisture-flux
uncertainty, on average, because the term (⳵D␩l /⳵b)
nonlinearly increases with growing relative volumetric
water content (Fig. 5). Since most modern soil models
are based on the Richards equation we must accept that
this uncertainty currently limits soil modeling in general. Therefore, an urgent need exists to determine
DECEMBER 2005
MÖLDERS ET AL.
1055
FIG. 7. Relationship between logarithm of soil-moisture fluxes
and logarithm of parameter-induced uncertainty in the fluxes at
various soil-temperature and relative volumetric water content
conditions. A difference in volumetric water content of 0.05 m3
m⫺3 over 0.05 m was assumed for calculations. Example shown for
sand.
more accurate values for b and develop an improved
parameterization for D␩l.
5) SOIL-HEAT
FLUX
At soil temperatures above 0°C, soil-heat fluxes decrease with diminishing total moisture nearly independent of soil temperature (Fig. 9). However, along the
freeze–thaw curve, Hs increases rapidly for decreasing
relative total moisture and first increases, then deFIG. 9. Soil-heat-flux uncertainty in dependence of (a) soil temperature and total volumetric water content for a gradient of 0.5
K over 0.05 m (example clay loam), and (b) soil temperature and
soil-temperature gradient.
FIG. 8. Horizontal distribution of soil-moisture flux (contour
lines) and its uncertainty (gray shading) in the second soil layer
(counted from the surface) after 45 h of simulation [kg (m2 s)⫺1].
creases as soil temperature lessens. At even lower temperatures, Hs decreases as temperature or relative total
moisture decrease.
For all soils, heat-flux uncertainty is of the same order of magnitude as typical errors in soil-heat flux measurements. Heat-flux uncertainty increases slightly with
increasing absolute value of soil-heat flux. The absolute
value of soil-heat flux and its relative error decreases
with increasing relative volumetric water content, indicating that Hs is more reliable after rain (e.g., Fig. 10) or
in the Tropics than during drought or in deserts.
Uncertainty is greater for great then small absolute
values of temperature gradients except if the flux approaches zero (e.g., Fig. 9). At soil temperatures below
0°C, heat-flux uncertainty exhibits a slight nonlinear
decrease with decreasing soil temperature for negative
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JOURNAL OF HYDROMETEOROLOGY
FIG. 10. As in Fig. 8, but for soil-heat flux (W m⫺2).
and positive temperature gradients (⳵Ts/⳵z) and is independent of soil temperature above 0°C. The steepness of the decrease (increase) depends on soil type.
Relative errors are maximal above the freeze–thaw
curve. In this temperature-moisture range, uncertainty
decreases nonlinearly toward colder and drier soil conditions.
On average, relative error increases in the following
order: sand, loamy sand, sandy loam, silty loam, loam,
sandy clay loam, silty clay loam, clay loam, sandy clay,
silty clay, clay, organic soils (Table 3). For some soil
temperature and soil-temperature gradient conditions
individual relative errors may exceed 50%.
For silt loam, sandy clay loam, silty clay loam, clay
loam, silty clay, and clay, {Hs, ␴b} exceed the contributions of other terms to heat-flux uncertainty by one or
two orders of magnitude, on average. In sand, {Hs, ␴␩s}
dominates heat-flux uncertainty (Table 2).
Since most modern soil models use the soil-heatdiffusion equation for Hs, individual terms derived for
Eq. (1) are analyzed. Excluding the cross effects [i.e.,
␳wL␷ D␩␷ (⳵␩/⳵z) ⫽ 0], which is frequently done to increase computational performance, shows that this
term does not cause great soil-heat-flux uncertainty.
The Dufour effect term leads to the behavior found for
dry-warm soil (Fig. 9). The examination also showed
that parameterization of thermal conductivity [Eq. (8)]
dominates heat-flux uncertainty, on average.
6) THERMAL
CONDUCTIVITY
Thermal conductivity ␭ as given by Eq. (8) depends
on b, ␺s, and ␩s, and shows nonlinear behavior (Fig. 11).
VOLUME 6
FIG. 11. Relationship between thermal conductivity given by
Eq. (8) and its uncertainty at various soil-temperature and relative
volumetric water content conditions. Example shown for loam.
For all soils, ␭ and its uncertainty less strongly decrease
with ␩/␩s below 0°C than above 0°C, and grow with
decreasing soil temperature. This increase is stronger
for high than low ␩/␩s. Thermal conductivity and its
uncertainty are temperature independent for soils
warmer than 0°C. On average, uncertainty is greatest
for loamy sand and least for clay, but still unacceptably
great (Table 3). Uncertainty of thermal conductivity is
greatest for wet hot soils (e.g., Tropics), and lowest for
dry frozen ground (e.g., midlatitudes winter). Note that
based on First International Satellite Land Surface Climatology Project (ISLSCP) Field Experiment (FIFE)
data, Peters-Lidard et al. (1998) reported underoverestimation of ␭ for wet/dry periods when Eq. (8) is
used.
Despite the huge relative errors of ␺s (␧␺s ⬎ 100%; cf.
Table 1), uncertainty of ␩s and b contribute more
strongly to soil-heat-flux standard deviation than does
that of ␺s (Table 2). Consequently, increasing the accuracy of ␩s and b provides more potential for improving soil modeling than increasing the accuracy of ␺s.
We modified Farouki’s (1981) approach, commonly
used in permafrost modeling, for application in
HTSVS,
␩ ␩ice 共␩s⫺␩⫺␩ice兲
␭ ⫽ ␭共s1⫺␩s兲␭w
␭ice ␭a
,
共12兲
and determined its parameter-induced statistical uncertainty. The only soil parameters occurring in this parameterization are thermal conductivity of dry soil material, ␭s, and porosity. Thermal conductivity of water
␭w[⫽0.57 W (m K)⫺1, ice ␭ice[⫽2.31 W (m K)⫺1], and
air ␭a[⫽0.025 W (m K)⫺1] are physical constants.
Although Eq. (12) provides lower thermal conduc-
DECEMBER 2005
MÖLDERS ET AL.
FIG. 12. Like Fig. 11, but for thermal conductivity given by Eq.
(12). Example shown for loam.
tivity values than Eq. (8), they fall in the range typically
measured in Alaska (V. E. Romanovsky 2004, personal
communication). Moreover, parameter-induced uncertainty in thermal conductivity is lower than when using
HTSVS’ original formulation and behaves less nonlinearly along the freeze–thaw curve (cf. Figs. 11 and 12).
The differences in soil-heat fluxes obtained with the
two parameterizations are of the same size as measurement errors. Thus, current measurements do not allow
assessing the superiority of one parameterization for
thermal conductivity over another. However, the
method illustrated here may serve as an objective measure for evaluating model improvement. As the parameter-induced uncertainty decreases when applying Eq.
(12), it is suggested this formulation be used in future
soil-model applications.
b. Application
Because moisture and heat fluxes are coupled [see
Eqs. (1)–(10)], precipitation can affect thermodynamic
state, particularly in the upper soil. Thus, the simulated
precipitation distribution is briefly described. For 21
July, MM5 predicts about 30- and 20-mm precipitation
in the Alaska and Brooks Ranges, respectively, and
heavy showers in western Alaska and the Interior, locally accumulating 7 mm. For 22 (23) July, precipitation
up to 7.5 (5) mm is predicted in the Brooks Range and
23 (36) mm in the Alaska Range. In the last 12 h of the
simulation, more than 22, 28, and 6 mm are predicted
for the Interior, Alaska Range, and Brooks Range, respectively. Traces of precipitation and local showers
occur in Yukon Territory and western Alaska.
The soil distribution, climatic, terrain, and soil-
1057
moisture conditions lead to the following picture:
Counted from the surface, on the North Slope, frozen
ground exists in the third layer and deeper, and in the
third or fourth layer in the Interior. In Yukon Territory,
Alaska Range, and toward the end of the episode in the
Brooks Range, frozen ground locally occurs at the surface. The deepest soil layer holds permafrost everywhere except along the Gulf of Alaska (Fig. 6).
Because statistical uncertainty in thermal conductivity propagates into soil-temperature and heat-flux uncertainty, a simulation was performed using Eq. (12).
This simulation provided slightly altered distributions
of Ts and Hs with, on average, lower uncertainties than
those of the simulation discussed in the following. As in
HTSVS, soil-temperature and soil-moisture states are
coupled; Ws, ␩, and hence their uncertainty changes
slightly too. These changes are small compared to the
reduced parameter-induced uncertainty found for Ts
and Hs. Note that slight changes in state variables and
fluxes slightly affected precipitation (onset, distribution, amount) due to altered water and energy fluxes to
the atmosphere. As no three-dimensional (3D) fields of
observed thermal conductivity exist, improvement of
MM5–HTSVS with the alternative parameterization
can only be evaluated indirectly, if at all, by assessing
overall performance in the future.
In the following, soil-heat and water fluxes having a
positive (negative) sign are directed toward the atmosphere (soil) and mean a cooling (heating) of the soil.
1) OVERVIEW
Typically, uncertainty and the contributions of {␾,
␴␹i} decrease with depth (up to more than an order of
magnitude) for all fluxes and state variables (e.g., Fig.
13) except at soil conditions above those characterized
by the freeze–thaw curve. Here, uncertainty in predicted Ts and ␩ reaches up to 6.1 K and 0.19 m3 m⫺3
when phase transitions occur. Great uncertainty exists
in areas experiencing a diurnal freeze–thaw cycle (e.g.,
Figs. 6 and 14). In deep soil, state variables and fluxes
and their uncertainties hardly change with time. Increased reliability at deeper layers results from lower
vertical temperature and moisture gradients and temporal changes in the permafrost soils. Note that some
studies (e.g., Bastidas et al. 2003; Yang et al. 2005)
found different results for soils in other areas; that is,
the uncertainty behavior with respect to depth cannot
be generalized. When applying inverse estimates of soil
properties, for instance, Yang et al. (2005) reported
greater errors for their deeper layers for a Tibet field
experiment. It has to be examined in the future whether
our findings are due to the fact that we assume the same
soil type at all depth.
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VOLUME 6
FIG. 13. Temporal evolution of uncertainty in (a) soil temperature, (b) volumetric water content, (c) soil-moisture fluxes, and
(d) soil-heat fluxes averaged over all land grid points at various depths. Results shown are after spinup.
In the 3D simulation, the soil-type dependency of the
flux–uncertainty relationship is less obvious than in the
theoretical analysis because the same flux value and
uncertainty can result from other temperature-moisture
combinations for different soils. At the surface, moreover, vegetation fraction and type indirectly influence
soil fluxes.
Uncertainty of state variables and fluxes shows a diurnal course in the uppermost soil layer, being most
evident for Hs and Ts (e.g., Fig. 13). For the given soil
conditions, uncertainty is usually greater by day than at
night. Precipitation strongly disturbs the diurnal cycle,
because percolating water increases moisture flux and
its induced heat transport can affect phase-transition
processes [cf. Eqs. (1) and (2)].
2) SOIL
TEMPERATURE
At low elevation, daytime soil-surface temperatures
reach up to 25°C in cloud-free, and 10°–20°C in cloudy,
areas; nighttime values vary between 5° and 15°C. In
the upper soil, temperatures are locally below 0°C in
the Brooks Range, in Yukon Territory, and the Alaska
Range (e.g., Fig. 14). In the deepest layer, temperatures
are below 0°C except along the Gulf of Alaska where
they may reach 5°C.
From the surface to as deep as the second soil layer
beneath the surface, soil-temperature uncertainty
shows a maximum around noon (e.g., Fig. 13). A secondary peak occurs around midnight in areas where
temperatures are above 0°C by day and lower at night.
In the temperature-moisture range above the freeze–
thaw curve (e.g., in the Brooks Range, Alaska Range,
Yukon Territory; see Fig. 14) temperature uncertainty
is the highest (⬎0.05 K) and contributions of {Ts, ␴␹i}
may reach 1 K. The highest temperature uncertainty
exists for sandy loam and loam areas because those soil
conditions fall in the range above the freeze–thaw
curve. Temperature uncertainty increases when rain
moistens the soil. When air temperatures slightly fall
during the episode, areas with frozen ground and consequently enhanced temperature uncertainty grow
slightly. In deep soil, uncertainty remains less than 10⫺3
K and distribution hardly varies with time.
At the surface, {Ts, ␴b}, {Ts, ␴␩s}, and {Ts, ␴␺s} each
contribute about 5 ⫻ 10⫺3 to 0.05 K to temperature
uncertainty except at some areas in Yukon Territory,
and the Alaska and Brooks Ranges. In the upper soil,
areas of clay loam are represented in the distribution of
{Ts, ␴␺s} by lower values (⬍5 ⫻ 10⫺3 K) than the adjacent soils.
DECEMBER 2005
FIG. 14. Horizontal distribution of soil temperature (lines) and
its uncertainty (shaded) in second soil layer (counted from the
surface) after 45 h of simulation.
3) SOIL
1059
MÖLDERS ET AL.
MOISTURE
In the upper soil layers of Interior and western
Alaska and most of Yukon Territory, ␩ ranges from
0.25 to 0.3 m3 m⫺3, whereas in the Brooks Range,
Alaska Range, southern Yukon Territory, and along
the Arctic Ocean, ␩ becomes less than 0.1 m3 m⫺3
because of permafrost (e.g., Fig. 6). At the surface,
evapotranspiration reduces ␩ slightly by day causing a
diurnal cycle in the upper soil (Fig. 13) where ␩ increases when precipitation occurs; ␩ drops right after
strong precipitation because of high saturated hydraulic
conductivity [cf. Eq. (6)]. Luo et al. (2003) found similar
behavior within PILPS. In deep soil, clay loam has low
ice content, and ␩ still varies between 0.15 and 0.2 m3
m⫺3. Along the Gulf of Alaska, ␩ ranges from 0.25 to
0.4 m3 m⫺3.
Surface-moisture uncertainty remains below 5 ⫻
10⫺4 m3 m⫺3 except for some locations in Yukon Territory, and for areas of precipitation. The lowest uncertainty exists in the Brooks and Alaska Ranges. In the
upper soil (e.g., Fig. 6), moisture uncertainty is two
orders of magnitude lower in clay-loam areas (⬍10⫺6
m3 m⫺3) than in their surroundings (⬍10⫺4 m3 m⫺3).
Surface-moisture uncertainty reaches up to 1.9 ⫻ 10⫺3
m3 m⫺3 during precipitation, on average. Once all water
has infiltrated, ␩ and its inherent uncertainty decrease
again.
Typically, no changes in response to atmospheric demands (water uptake by roots for transpiration, upward
soil-moisture fluxes in response to evaporation) can be
detected in moisture uncertainty beneath the uppermost layer except for clay and clay loam. The higher
tolerance of these soil types for coexistence of ice and
supercooled water (Fig. 2) explains why atmospheric
demands can affect deeper soil layers. Thus, diurnal
temperature variations cause fewer changes in partitioning between solid and liquid phases in all clay soils
than in other soils.
In loam and sandy loam, maximum relative supercooled water content decreases rapidly as temperatures
decrease during our episode. Thus, appreciable uncertainty exists in these areas. In deeper soil, moisture
uncertainty remains highest in loam and smallest in clay
and clay loam, but relative errors are small. In the deepest soil layer, values are negligible (⬍10⫺8 m3 m⫺3) in
areas of continuous (Yukon Territory, Alaska Range,
Brooks Range, North Slope) and discontinuous (⬍10⫺7
m3 m⫺3; Interior, western Alaska) permafrost.
Of all hydraulic parameters, uncertainty in ks contributes the least (⬍10⫺5 m3 m⫺3 nearly everywhere) to
moisture uncertainty in the upper soil. In the Alaska
Range, Brooks Range, parts of Yukon Territory, along
the Arctic Ocean, and in clay loam, {␩, ␩s} remains
negligible in the upper soil. Elsewhere this term
amounts to between 5 ⫻ 10⫺6 to 5 ⫻ 10⫺4 and 5 ⫻ 10⫺5
to 10⫺4 m3 m⫺3 in areas without and with precipitation,
respectively. In the upper soil of the Alaska Range,
Yukon Territory, Interior, and along the Gulf of
Alaska, {␩, ␴b} ranges between 2.5 ⫻ 10⫺5 and 7.5 ⫻
10⫺5 m3 m⫺3; it is less elsewhere, and even negligible on
the North Slope, in the Brooks Range, and in parts of
the Alaska Range. In the upper soil, contributions by
{␩, ␴␺s} stay below 10⫺4 m3 m⫺3 nearly everywhere, and
are negligible in the Brooks Range, Alaska Range,
parts of Yukon Territory, and in clay loam.
4) SOIL-MOISTURE
FLUX
By day, upward soil-moisture fluxes [up to 10⫺4 kg
(m2 s)⫺1] occur almost everywhere at the surface except
along the Gulf of Alaska and at some locations in Interior and western Alaska and Yukon Territory. At
night, surface soil-moisture fluxes remain positive north
of the Arctic Circle because here evapotranspiration
still occurs. Here, Ws penetrates upward all day in the
uppermost soil layer to fulfill atmospheric demands
(e.g., Fig. 8). Soil-moisture fluxes bear more statistical
uncertainty in cloudless than in rain-free cloudy areas,
because in the former evapotranspiration is usually
higher than in the latter. Moisture-flux uncertainty is an
order of magnitude greater during and just after precipitation than before precipitation onset. Precipitation
impact on moisture-flux uncertainty decreases with
depth and after precipitation, as Ws decreases. In the
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JOURNAL OF HYDROMETEOROLOGY
second soil layer beneath the surface, soil-moisture
fluxes are less than 10⫺4 kg (m2 s)⫺1 (e.g., Fig. 8). Here,
they hardly show a diurnal cycle, but still respond
slightly to precipitation. In deeper soil layers, Ws becomes negligible in permafrost and small in unfrozen
soil.
At the surface, moisture-flux uncertainty remains below 3.4 ⫻ 10⫺3 kg (m2 s)⫺1, on average (Fig. 13), with
the greatest uncertainty for loamy sand, and the smallest for clay loam and clay [⬍10⫺4 kg (m2 s)⫺1]. Note
that the former areas received precipitation locally,
while the latter did not.
Close beneath the surface (e.g., Fig. 8), clay loam
areas are represented in the distribution of moistureflux uncertainty by lower values [⬍5 ⫻ 10⫺6 kg (m2 s)⫺1]
as compared to their surroundings [2.5 ⫻ 10⫺5–5 ⫻
10⫺5 kg (m2 s)⫺1]. In the third layer beneath the surface, sandy loam areas are indicated by higher uncertainty values [up to 10⫺4 kg (m2 s)⫺1] than adjacent soils
[⬍10⫺5 kg (m2 s)⫺1]. At deep layers, moisture-flux uncertainty is less than 10⫺7 kg (m2 s)⫺1 in permafrost and
less than 10⫺5 kg (m2 s)⫺1 elsewhere.
Along the Gulf of Alaska and in Interior and western
Alaska, {Ws, ␴b} ranges from 5 ⫻ 10⫺6 to 10⫺5 kg (m2 s)⫺1.
In permafrost, this term contributes negligibly. Areas of
sandy loam are clearly indicated in the distribution of
{Ws, ␴␺s} by values lower [⬍2.5 ⫻ 10⫺5 kg (m2 s)⫺1]
than those in the environment [2.5 ⫻ 10⫺5–5 ⫻ 10⫺5 kg
(m2 s)⫺1]. The same is true for clay loam in deeper soil
[⬍10⫺7 kg (m2 s)⫺1]. Uncertainty in ks contributes less
to moisture-flux uncertainty than all other parameters.
In deep soil, clay loam is clearly represented in the {Ws,
␴ks} distribution by smaller values [⬍10⫺7 kg (m2 s)⫺1]
compared to adjacent soils.
5) SOIL-HEAT
FLUX
During the episode, soils cool in Interior and western
Alaska, in most of Yukon Territory, and in the western
North Slope at night. Daytime soil heating is greatest in
cloud-free areas, locally at the surface as high as 150 W
m⫺2 (Yukon Territory). In cloudy areas, heating ranges
from 25 to 50 W m⫺2. At night, Hs varies between ⫺25
and 50 W m⫺2. In some areas, Hs does not change
direction in the diurnal course. Soil-heat fluxes decrease with distance from the surface in the upper soil,
but still heat the soil during the day. At deeper layers,
of course, Hs hardly varies with time.
By day, heat-flux uncertainty reaches up to 50 W m⫺2
in the Brooks and Alaska Ranges and in cloud-free
areas in Yukon Territory, while it remains less than 30
W m⫺2 elsewhere (e.g., Fig. 10). At night, uncertainty
reduces to less than 10 W m⫺2 except for cloud-free
areas in western Alaska and southern Yukon Territory
VOLUME 6
where it remains below 50 W m⫺2. The lowest heat-flux
uncertainty exists in permafrost (e.g., Brooks Range,
Alaska Range, locally in Yukon Territory). At deeper
layers, uncertainty typically reduces to 5 (20) W m⫺2 in
areas with (without) permafrost.
At the surface and in the upper soil, heat-flux uncertainty shows a maximum around noon (Fig. 13). If Hs
changes direction, uncertainty shows a secondary peak
at midnight. Since at this time of year absolute soilheat-flux values are greater by day than at night, uncertainty of downward soil-heat fluxes exceeds that of
upward fluxes. In the third layer, diurnal amplitude still
reaches up to 10 W m⫺2, on average.
At soil conditions above those represented by
the freeze–thaw curve, {Hs, ␴␺s}, {Hs, ␴␩s}, and {Hs, ␴b}
each typically exceed 20 W m⫺2. In deep soil, {Hs, ␴␺s}
usually remains less than 2 W m⫺2. In permafrost, {Hs,
␴␺s}, {Hs, ␴␩s}, and {Hs, ␴b} stay below 0.5 W m⫺2.
5. Conclusions
We introduced GEP principles to examine model uncertainty in predicted values of ␩, Ts, Ws, and Hs caused
by statistical uncertainty of empirical soil parameters
occurring in the governing equations of the soil physical
processes. This method allows examining the relative
importance of these parameters in producing forecast
uncertainty at various forecast lead ends. Close beneath
the surface, uncertainty undergoes a diurnal course. At
most soil temperature-moisture conditions, a quasilinear relationship exists between the absolute values of
Ws and Hs and their respective uncertainty; that is, relative predictability is about as good by day as at night.
When phase transitions occur, the freeze–thaw term
causes great uncertainty in ␩ and Ts when compared to
the quantities themselves. This explains the great uncertainty throughout the simulation at temperaturemoisture conditions above the freeze–thaw curve. Thus,
we may conclude that predictions of ␩ and Ts are least
reliable in the active layer close to the freezing line and
in moist soils in midlatitude winter as temperature falls.
Our analysis gave evidence that uncertainty in thermal conductivity dominates heat-flux uncertainty. Introducing Farouki’s (1981) formulation of thermal conductivity in HTSVS reduced uncertainty in predicted
values of Hs and Ts especially above the freeze–thaw
curve. These findings underline that GEP principles are
indispensable for analysis of parameterized soil processes. As documented in Tables 2 and 3, our analysis
also identified predictions for soils with high clay fraction as especially uncertain.
Pore-size distribution index was identified as the
most critical parameter. Its uncertainty especially domi-
DECEMBER 2005
MÖLDERS ET AL.
nates uncertainty in Ws because it causes high relative
errors in Kw. If uncertainty were judged according to
the parameters’ relative errors, ks followed by b would
be classified as the parameters for which higher accuracy would be desirable. Uncertainty in thermal parameters generally contributes less to soil-temperature uncertainty than uncertainty in hydraulic parameters.
Even for Hs, increased accuracy of hydraulic parameters will reduce uncertainty because parameterization
of thermal conductivity depends on hydraulic parameters. Since increasing the accuracy of hydraulic parameters offers a greater potential for improvement of soil
modeling than doing so for thermal parameters, efforts
should focus on achieving the former.
Since GEP showed itself able to identify critical parameters and (parts of) parameterizations, results of
GEP analysis could form a basis for prioritizing which
parameters to determine with higher accuracy and for
intercomparisons of soil models aimed at improving soil
modeling.
Acknowledgments. We thank the reviewers for helpful discussion and fruitful comments, C. O’Connor for
editing, and BMBF and NSF for financial support under contracts 07ATF30, ATM-0232198, and OPP0327664.
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